Is there any way to move/deform procedural work along a mesh as a normal texture would?
It was the lattice!

As you probably know, procedurals need some coordinate space to originate, and it can be something's origin or the world's point zero. In this case the nodes setup could generate a perfectly fine texture that could move along with the mesh as a UV, but since procedurals aren't really a texture but a "projection" (can't think of a better definition) they move with the mesh's coordinates, but not with the vertexes. And this means that as long there's no deformation on the mesh everything is fine... But the lattice was deforming the mesh...

Hope you find it useful if you're started on procedurals. For this, coordinate spaces are really important. I came up with a metaphor about it:

Think of procedurals as a beam of light made of x-rays that need something else's origin as it's emitter. As it reaches the mesh's surface, it illuminate the surface the way it wants then go right through the object and keep going, because it's x-ray. Changing the surface won't change the beam of light